Topological Approach to Microcanonical Thermodynamics and Phase Transition of Interacting Classical Spins

We propose a topological approach suitable to establish a connection between thermodynamics and topology in the microcanonical ensemble. Indeed, we report on results that point to the possibility of describing {\it interacting classical spin systems} in the thermodynamic limit, including the occurrence of a phase transition, using topology arguments only. Our approach relies on Morse theory, through the determination of the critical points of the potential energy, which is the proper Morse function. Our main finding is to show that, in the context of the studied classical models, the Euler characteristic $\chi(E)$ embeds the necessary features for a correct description of several magnetic thermodynamic quantities of the systems, such as the magnetization, correlation function, susceptibility, and critical temperature. Despite the classical nature of the studied models, such quantities are those that do not violate the laws of thermodynamics [with the proviso that Van der Waals loop states are mean field (MF) artifacts]. We also discuss the subtle connection between our approach using the Euler entropy, defined by the logarithm of the modulus of $\chi(E)$ per site, and that using the {\it Boltzmann} microcanonical entropy. Moreover, the results suggest that the loss of regularity in the Morse function is associated with the occurrence of unstable and metastable thermodynamic solutions in the MF case. The reliability of our approach is tested in two exactly soluble systems: the infinite-range and the short-range $XY$ models in the presence of a magnetic field. In particular, we confirm that the topological hypothesis holds for both the infinite-range ($T_c \neq 0$) and the short-range ($T_c = 0$) $XY$ models. Further studies are very desirable in order to clarify the extension of the validity of our proposal

Análise estrutural da vegetação sobre um afloramento rochoso granítico de altitude no Parque Estadual da Serra do Brigadeiro, Minas Gerais, Brasil

Granitic rock outcrops of the Brazilian southeast are either coastal or inland. The latter can often have high altitudes, such as in the summits of Serra do Mar and Serra da Mantiqueira, where they are known as Campos de Altitude. The landscape on these high altitude plateaux is often a mosaic of shrubs and treelets within a bunchgrass matrix, with sparse pteridophytes and other herbs, interspersed with variable extensions of rock outcrops. Despite the pervasiveness of rock outcrops in the Brazilian landscape, studies on the structural analysis of the vegetation on such formations are scarce. This study aimed to analyze the structure of the vegetation on a highland granitic rock outcrop in Parque Estadual da Serra do Brigadeiro, state of Minas Gerais, Southeast Brazil (42°20' - 42°40' S and 20°20' - 21°00' W, 1,722 m of elevation ). Quantitative parameters of absolute and relative frequency and dominance (cover) were estimated. The group analysis used the Jaccard similarity index. Trilepis lhotzkiana, Panicum sp. 1, and Vellozia variegata presented the highest relative frequencies, relative dominances and importance values. These three species, along with Dyckia bracteata, Rhynchospora emaciate, and Tibouchina cf. manicata, represented 98.3% of the relative dominance. The remaining 1.7% referred to 22 remaining species. The distinction among quadrats within formed groups by cluster analysis was due to the great number of low frequency species.Os afloramentos rochosos graníticos do sudeste brasileiro ocorrem no litoral ou no interior. Neste segundo caso, incluem-se os Campos de Altitude, como aqueles encontrados nos cumes das Serras do Mar e da Mantiqueira. A fisionomia mais freqüentemente encontrada nos platôs é composta de mosaicos de arbustos e arvoretas, inseridos em uma matriz de touceiras de gramíneas, com pteridófitas e outras ervas dispostas espaçadamente, além de extensões variáveis de afloramentos rochosos. Apesar de freqüentes na paisagem brasileira, são escassos os trabalhos de análise estrutural da vegetação sobre afloramentos rochosos. O objetivo deste trabalho foi analisar a estrutura da vegetação sobre um afloramento rochoso granítico de altitude no Parque Estadual da Serra do Brigadeiro, MG (42°20' - 42°40' S e 20°20' - 21°00' W, 1.722 m de altitude). Foram estimados os parâmetros quantitativos de freqüência e dominância (cobertura), absolutas e relativas. A análise de agrupamento utilizou o índice de similaridade do Jaccard. Trilepis lhotzchiana, Panicum sp. 1 e Vellozia variegata apresentaram as maiores freqüências relativas, dominâncias relativas e os maiores valores de importância. Essas três espécies mais Dyckia bracteata, Rhynchospora emaciata e Tibouchina cf. manicata responderam por 98,3% da dominância relativa, os restantes 1,7% ficaram distribuídos por outras 22 espécies. A distinção das parcelas dentro dos grupos, formados na análise de agrupamento foi atribuída ao grande número de espécies de baixa freqüência.657664Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq